3.621 \(\int x^5 (a+b x^4)^2 \, dx\)

Optimal. Leaf size=30 \[ \frac {a^2 x^6}{6}+\frac {1}{5} a b x^{10}+\frac {b^2 x^{14}}{14} \]

[Out]

1/6*a^2*x^6+1/5*a*b*x^10+1/14*b^2*x^14

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Rubi [A]  time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {270} \[ \frac {a^2 x^6}{6}+\frac {1}{5} a b x^{10}+\frac {b^2 x^{14}}{14} \]

Antiderivative was successfully verified.

[In]

Int[x^5*(a + b*x^4)^2,x]

[Out]

(a^2*x^6)/6 + (a*b*x^10)/5 + (b^2*x^14)/14

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int x^5 \left (a+b x^4\right )^2 \, dx &=\int \left (a^2 x^5+2 a b x^9+b^2 x^{13}\right ) \, dx\\ &=\frac {a^2 x^6}{6}+\frac {1}{5} a b x^{10}+\frac {b^2 x^{14}}{14}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 30, normalized size = 1.00 \[ \frac {a^2 x^6}{6}+\frac {1}{5} a b x^{10}+\frac {b^2 x^{14}}{14} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*(a + b*x^4)^2,x]

[Out]

(a^2*x^6)/6 + (a*b*x^10)/5 + (b^2*x^14)/14

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fricas [A]  time = 0.53, size = 24, normalized size = 0.80 \[ \frac {1}{14} x^{14} b^{2} + \frac {1}{5} x^{10} b a + \frac {1}{6} x^{6} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x^4+a)^2,x, algorithm="fricas")

[Out]

1/14*x^14*b^2 + 1/5*x^10*b*a + 1/6*x^6*a^2

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giac [A]  time = 0.15, size = 24, normalized size = 0.80 \[ \frac {1}{14} \, b^{2} x^{14} + \frac {1}{5} \, a b x^{10} + \frac {1}{6} \, a^{2} x^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x^4+a)^2,x, algorithm="giac")

[Out]

1/14*b^2*x^14 + 1/5*a*b*x^10 + 1/6*a^2*x^6

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maple [A]  time = 0.00, size = 25, normalized size = 0.83 \[ \frac {1}{14} b^{2} x^{14}+\frac {1}{5} a b \,x^{10}+\frac {1}{6} a^{2} x^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(b*x^4+a)^2,x)

[Out]

1/6*a^2*x^6+1/5*a*b*x^10+1/14*b^2*x^14

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maxima [A]  time = 1.35, size = 24, normalized size = 0.80 \[ \frac {1}{14} \, b^{2} x^{14} + \frac {1}{5} \, a b x^{10} + \frac {1}{6} \, a^{2} x^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x^4+a)^2,x, algorithm="maxima")

[Out]

1/14*b^2*x^14 + 1/5*a*b*x^10 + 1/6*a^2*x^6

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mupad [B]  time = 0.03, size = 24, normalized size = 0.80 \[ \frac {a^2\,x^6}{6}+\frac {a\,b\,x^{10}}{5}+\frac {b^2\,x^{14}}{14} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a + b*x^4)^2,x)

[Out]

(a^2*x^6)/6 + (b^2*x^14)/14 + (a*b*x^10)/5

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sympy [A]  time = 0.10, size = 24, normalized size = 0.80 \[ \frac {a^{2} x^{6}}{6} + \frac {a b x^{10}}{5} + \frac {b^{2} x^{14}}{14} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(b*x**4+a)**2,x)

[Out]

a**2*x**6/6 + a*b*x**10/5 + b**2*x**14/14

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